The essential minimum of the height $h_{\overline{L}}$ is defined as

$$e(\overline{L})=\sup_{\begin{subarray}{c}\emptyset\neq U\subset X\end{subarray}}\inf_{P\in U(\overline{F})}h_{\overline{L}}(P),$$

where the supremum runs over non-empty open subsets of $X$. If $L$ is big, then $e(\overline{L})$ is a real number. Another way to state its definition is the following : for any real number $B$, then the set

$$\{P\in X(\overline{F})\,;\,h_{\overline{L}}(P)\leq B\}$$

is Zariski dense if $B>e(\overline{L})$, and is not Zariski dense if $B<e(\overline{L})$.

Assume that $\overline{L}$ is an ample line bundle on $X$, equipped with a semi-positive adelic metric. The (geometric/arithmetic) Hilbert-Samuel theorem implies the following inequality

$$e(\overline{L})\geq\frac{({\widehat{c}}_{1}(\overline{L})^{n+1}|X)}{(n+1)(c_{1}(L)^{n}|X)}.$$

(See Zhang [59], as well as [38, 28] for more details in the geometric case). When $X$ is a curve and $F$ is a number field, Autissier [3] proved that the inequality holds for any ample line bundle with an admissible adelic metric (see [18]) ; this extends to the geometric case.